Note that we do not consider permanent price impact in our formulation. Within the
continuous-time Almgren–Chriss framework, it is well known that the contribution of permanent
linear price impact to the implementation shortfall is path-independent; i.e., it does not depend
on the liquidation strategy as long as the strategy clears all the positions eventually.Suppose that liquidating the asset at rate \(v\)  permanently shifts the market price at rate
\(-\lambda v\) , i.e., permanent linear price impact. Given a feasible policy \(\pi \) , the
associated transaction costs can be expressed \(\int _{t=0}^\infty \lambda \pi _t X_t^{x,\pi } dt\) ,
which is always equal to \(\frac{1}{2} \lambda x^2\) , independent of the policy, given that
\(\lim _{t \rightarrow \infty } X_t^{x,\pi } = 0\) . Therefore, without loss of generality, we can
ignore the presence of permanent impact since it does not make any difference in the trader's
decision making. See [1]} for details.
